8 The Structure of Collagen 157Fig. 8.9.Two successive twist grain boundaries overlap–gap–overlap. This illus-
trates the 5/4 ratio between the areas of the overlap and gap unit cells. The
structure of a collagen fibril is thus a periodic stacking of twist grain boundaries
[overlap−gap] 10
8.5 Transverse Structures (gap, overlap) on Two
Orthogonal Triangular Lattices
Since the collagen helix is represented on a cylinder covered by a triangular
lattice (Fig. 8.2c and 8.3), and since it is inflated by 1 +
√
3, the transverse
structure of the collagen fibril should be based on an underlying triangular
lattice.
The transverse structure is, topologically, a square–triangle pattern
32. 4. 3 .4, with vertex coordinationz= 5. This pattern accommodates the flex-
ibility of tightly packed individual molecules through gaps in their stacking
(Fig. 8.1d). Accordingly, the transverse structure at the “gap” level is topolog-
ically the same as at the “overlap” level, but with density reduced by 4/5, and
overall rotation (see Fig. 8.5b).^7 The transition overlap–gap–overlap is a twist
grain boundary (TGB). In general, the one-dimensional stacking of TGB is
periodic [9], but, in collagen, this periodicity, and the square–triangle trans-
verse structure, are necessary consequence of close packing of amino acids,
that is of steric repulsion.
The mechanism of the transition from overlap to gap is illustrated in
Fig. 8.5 [10], showing how an interruption in one out of every five molecules
leaves the square–triangle lattice invariant. The topological rotation is by
tan−^1 (1/2). Figure 8.5 also shows that the square–triangle (z= 5) and square
(z= 4) patterns are the only topological lattices invariant through regularly
spaced gaps. Each vertex has a label 1, 2 , 3 , 4 ,5 (mod.5) that indicates the gap
level at which its representative molecule is interrupted. The numbers increase
by 3 (mod.5) horizontally, and by 1 vertically, in the positive sense (by 2 and
by 4 in the negative sense). The interrupted molecule in the gap region has
four neighbours with labels all different. Every vertex in the gap region has
five neighbours, with labels different from its own and from that of the gap.
(^7) The fact that out of the five molecules in the overlap, only four extend into the
gap has been established by Hodge and Petruska [8]. See [3, 10].